Introduction to Quantum Field Theory
Quantum Field Theory is a Relativistic Quantum Mechanics, but with fields instead of finitely many particle co-ordinates. The motivation comes from the principles of Locality and Lorentz-Invariance. Non-relativistic Quantum Field Theoryies exist, however, they are not Lorentz-Invariant. Quantum Field Theory shares many of the properties and equations, of Relativistic Quantum Mechanics; however, they are not applied to Wavefunctions, but Wave Functionals instead. Importance Quantum Mechanics is not Lorentz-Invariant, while Special Relativity is Classical. Quantum Field Theory "rederives" Quantum Mechanics, but using special relativistic, as compared to non-relativistic Lagrangians, etc. This is not as simple as it may seem since even the simple Klein-Gordon Equation involves a simple trick in it's derivation,. Also, the Klein-Gordon Equation only applies as a field equation for particles with 0 spin, e.g. the Higgs Boson. One needs instead the Dirac Equation, which is like the "square root" of the Klein-Gordon Equation, and it is derived by factoring the Klein-Gordon Equation. Even with the Dirac Equation, certain paradoxical results still appear to remain, and it is still the realm of Relativistic Quantum Mechanics. To progress to Quantum Field Theory, one needs to apply the principle of Locality, and this forces us to employ Quantum Fields. Overview Relativistic Quantum Mechanics Quantum Mechanics is by default, not Lorentz-Invariant. In Relativistic Quantum Mechanics, one starts with a Special Relativistic Lagrangian Density, which eventually obtains the Klein-Gordon Equation. However, this result does not have a conserved non-negative probability current. Therefore, one derives the Dirac Equation, which however continues to not have a conserved non-negative probability current. Quantum Fields The solution to the issue with Relativistic Quantum Mechanics involves the use of Quantum Fields, which are spinor-valued. The wavefunctions are replaced by wave functionals. This solves the problem with the Probability Densityies not being conserved. In this new formulation, the Dirac Equation is a field equation for spin-1/2 fields, i.e. Dirac Fields, whereas the Klein-Gordon Equation is a field equation for spin-0 fields, i.e. Klein-Gordon Fields. Renormalisation It is common to see divergent perturbation series in quantum field theory. Renormalisation is a technique to handle such results by introducing a cut-off \Lambda ; it is possible to show, though, that physical results, such as scattering amplitudes, are cutoff-independent. The Wilsonian philosophy of renormalisation explains that Quantum Field Theories are only relevant upto this cut-off, and there is a deeper, general, underlying theory. The Standard Model It can be observed that the Lagrangian Densityies of Quantum Chromodynamics, Electroweak Theory, and the Higgs Mechanism, are consistent with each other. They could then be combined into a single Lagrangian Density. Of course, this is man-made, and suited to experimental predictions, and therefore a model. Semi-Classical Gravity The Standard Model is a Special Relativistic model, however, it is not General Relativistic, and does not explain gravity. Semi-Classical Gravity is an attempt to explain gravity through Quantum Field Theory, by working the Standard Model on curved spacetime; however, it explains gravity classically, while matter and the other fields quantumly, which makes it inconsistent, as can be seen below: The Einstein-Field Equation itself, in a certain system of natural units, is written as follows: G_{\mu\nu}=T_{\mu\nu} There are clearly no expectation value signs. However, in Semi-Classical Gravity, since matter is quantum, and gravity is classical, this would be inconsistent, and one would need to employ expectation values: G_{\mu\nu}=\langle T_{\mu\nu} \rangle Which is inconsistent with pure General Relativity. A true quantum theory that incorporates gravity is expected to have the same sort of field equation, with the Einstein Tensor and Stress-Energy-Momentum Tensors themselves, quantum fields. Issues with the Standard Model In spite of the success of the Standard Model, it has a few major issues: * It is incapable of explaining gravity in a quantum, consistent manner. * It is a model, and therefore it is designed to fit experiments. It however, is incapable of explaining all experiments. * It has 19 free parameters, which are adjusted arbitrarily to match experiment. * It is a perturbative theory (and employs renormalisation for that reason), so one needs to discover the more fundamental non-perturbative version. Beyond the Standard Model As previously seen, the Standard Model is not truly a complete theory. Theories that attempt to solve one or more of these problems are known as Beyond The Standard Model (BSM) theories. BSM theories are classified into Theories of Everything (ToE) and Grand Unified Theories (GUT). Grand Unified Theories Grand Unified Theories unify the three Standard Model gauge forces of Electromagnetism, the Weak Force, and the Strong Force into a simple gauge group which can be broken down into the Standard Model gauge group. There is no gravity in these theories. Theories of Everything A Theory of Everything is a more ambitious goal as compared to a Grand Unified Theory. A Theory of Everything is a theory that unifies gravity with the gauge forces of the Standard Model, and also quantises gravity at the same time. In a way, it combines the goals of a theory of Quantum Gravity and that of a Grand Unified Theory. String Theory is an example of an aspirant Theory of Everything. History Experimental Tests Applications